Let $g$ be a complex simple Lie algebra and $U_q(g)$ the corresponding quantum group.
Is the category of all finite dimensional modules of $g$ equivalent to the category of all finite dimensional modules of $U_q(g)$?
It seems that when $q$ is a root of unity, the category of all finite dimensional modules of $g$ is equivalent to the category of all finite dimensional modules of $U_q(g)$. When $q$ is not a root of unity, the category of all finite dimensional modules of $g$ is not equivalent to the category of all finite dimensional modules of $U_q(g)$. Is this true? Thank you very much.